Asymptotics for products of characteristic polynomials in classical $\beta$-Ensembles

Abstract

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles of $N\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as $N\to \infty$. In the bulk of the spectrum of each $\beta$-ensemble, the same scaling limit is found to be $e^{p_1}{}_1{F}_1$ whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre $\beta$-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when $\beta$ is even, scaling limits of the $k$-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson's lemma and the steepest descent method for integrals of Selberg type.

[v3] 35 pages; this is a revised and enlarged version of the article with new references, simplified demonstations, and improved presentation. To be published in Constructive Approximation 37 (2013)

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• From Random Matrices to Stochastic Operators
• Matrix model version of AGT conjecture and generalized Selberg integrals
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• Diffusion at the random matrix hard edge
• Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures
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• Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials
• Edge scaling of the β-Jacobi ensemble

Cited by in corpus (5)

• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Loop Equation Analysis of the Circular $\beta$ Ensembles
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles
• Phase transitions for products of characteristic polynomials under Dyson Brownian motion